A nonlinear optics equation with transformations of the independent space variable as controlling actions
A mathematical modelling of nonlinear optical systems with nonlocal field transformations in a feedback outline is discussed. A differential equation of the form \[ \tau u_t+{\mathbf L}u=\alpha\cos u,\quad t \geq 0,\;0 < x < 2\pi,\tag{1} \] where \(u=u(t,x)\) is a phase modulation of light beams, \(\tau>0\) is the time of relaxation of a nonlinear response, \({\mathbf L}u=u-{\mathbf D}u_{xx}\), \({\mathbf D}>0\) is a diffusion coefficient, \(\alpha, \beta \) are parameters, is taken as a base system. The differential equation (1) describes the phase modulations \(u(t,x)\) of the light beam in a system with nonlinear optical elements. The values of the parameters \(\alpha > 0\), \(\beta > 0\) are defined by an intensity of a space-homogeneous input emission and by dissipative characteristics of the feedback outline. The periodic boundary conditions \[ u(t,0)=u(t,2\pi),\;u_x(t,0)=u_x(t,2\pi),\;t>0,\tag{2} \] and initial conditions \[ u(0,x)=u_0(x),\;0<x<2\pi,\tag{3} \] are additionally given to (1). The question of existence, uniqueness and regularity of a generalized solution of (1) with the boundary and the initial conditions (2), (3) are examined in the article.
scientific article; zbMATH DE number 1407639